Hrm. That's really difficult. I mean, I do have a degree in history and there are so many times and places where I would love to go back and be a fly on the wall. Or, better, hide some small cameras and/or audio bugs around key places and times. (As long as, you know, I didn't have to actually stay there.) If I were doing it for personal interest ... maybe the time of Christ, and follow him around?
You see, we have no historical documents about Jesus or any of his disciples. None. We have the Gospels and Acts, of course, but those are not history books and were never intended to be. (Even by the standards of the day, much less our standards.) This is what people who read the Bible today pretty much always miss. While the Bible tells true stories, that doesn't mean it tells factual ones.
Look, when a historian today works, they gather all the facts, sift through them, and try to figure out what they mean--to figure out, in other words, the truth. And if they get any of their facts wrong, they get ripped to shreds. But in ancient times, historians worked differently. They figured out what the truth was, and then figured out how best to arrange the facts and garnish them so as to help people understand that truth. Which is how you get things like Josephus giving us what he says is the speech given in the fortress of Masada the night before its inhabitants committed suicide to the last man. Either he "improved" things and there were survivors to tell him about the speech, or he made up the speech because he thought it would be the most interesting way of conveying to his listeners the ideals the Jewish rebels at Masada were fighting for. So even when you're reading history from the ancient world (or, in fact, any place prior to the Enlightenment, or any place where Western scientific theory hasn't come to dominate academia) you have to take that into account. They told history like a story and would sometimes alter/embellish the facts to fit or dramatize things.
But even by ancient standards, the Gospels are not history books. (There are history books in the Bible--Kings, Chronicles, Samuel--but they give very different pictures of the same events, and are definitely of the ancient model, which is not what we modern Western people expect history to be.) The Gospels are "gospels," in Greek "euangelions"--and if that looks familiar, it should, it's the word that "evangelism" comes from and literally it means "good news." The Gospels are designed to teach people the good news that Jesus came to bring through stories about Jesus' life. The theology is the important part, not the history. Which is why, for example, different Gospels record that Jesus was crucified on different days.* Jesus' death was near Passover and theologically connected to it, but different Gospels explained that connection in different ways, resulting in different days for the crucifixion. The theological point was more important than factual accuracy. And more than that: the effect of that theological point on the reader or hearer was the most important thing.
All of which means that (despite all the ink spilled on the subject) there is very little we know for sure about Jesus and his disciples, from a factual historical point of view. I've always been curious as to what actually happened, but it's not a matter of faith for me--that is, if I went back and found that things were very different from the way the Gospels tell the story, I doubt it would affect my faith because I don't read the Gospels for historical fact in the first place.
*Matthew, Mark, and Luke (the Synoptic Gospels) record that Jesus' last meal was a Passover meal (eaten on Passover Eve), and that he died the next day (still Passover, because the Jewish day starts at sunset). John, however, records that Jesus' last meal was an ordinary friendship meal the day before Passover Eve, and that Jesus died on the Day of Preparation (i.e. he died while the Passover lambs were being slaughtered in the Temple). John wants to hammer home that Jesus is the Passover Lamb, the Lamb of God, the one who dies to save the people from the angel of death, so he dies as the lamb dies. The Synoptics want to hammer home that the Lord's Supper is connected to the Passover Meal, eaten the last night before the slaves become free, and Jesus' body and blood (in the form of bread and wine) are like the blood of the passover lamb, even though he doesn't die until later. All four make similar connections, but there are different shades of meaning. People argue: which day did Jesus die? (After all, he can't have died on both days, and he can't have had two Last Suppers.) I don't think it matters. I think that if the writers of the Gospels knew people were arguing about it and thought it was a major deal, they would have been shocked and horrified. But with our modern fact-based educations, that's the sort of thing we focus on.